A061718 -id:A061718 - cp's OEIS frontend

A085741 a(n) = T(n)^n, where T() are the triangular numbers (A000217).

1, 1, 9, 216, 10000, 759375, 85766121, 13492928512, 2821109907456, 756680642578125, 253295162119140625, 103510234140112521216, 50714860157241037295616, 29345269354638035222576971

Offset: 0

Jon Perry, Jul 21 2003

			a(3) = T(3)^3 = 6^3 = 216.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000217.
Essentially the same as A061718.
Cf. A000312, A062206, A212333.

[((n*(n+1))/2)^n: n in [0..20]]; // Vincenzo Librandi, Sep 14 2011

a:=n->mul(sum(j, j=1..n),k=1..n): seq(a(n), n=0..13); # Zerinvary Lajos, Jun 02 2007

With[{rnn=Range[20]},Join[{1},First[#]^Last[#]&/@Thread[ {Accumulate[ rnn],  rnn}]]] (* Harvey P. Dale, Dec 08 2013 *)

a(n) = (n*(n+1)/2)^n; \\ Michel Marcus, Feb 19 2019

a(n) = ((n*(n+1))/2)^n. - Vincenzo Librandi, Sep 14 2011

More terms from Ray Chandler, Nov 09 2003

A070307 Number of n X n matrices with nonnegative integer entries such that every row sum equals 3.

Original entry on oeis.org

1, 16, 1000, 160000, 52521875, 30840979456, 29509034655744, 42998169600000000, 90647430472564453125, 265599227914240000000000, 1047192117300356121695451136, 5410240907043328777415185924096, 35821862005173382840059779052734375, 298285661929377847941529600000000000000

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 10 2002

nonn

Indranil Ghosh, Table of n, a(n) for n = 1..160

Cf. A061718, A000292.

Table[ Binomial[n + 2, 3]^n, {n, 1, 14}]

a(n) = binomial(n+2, 3)^n; \\ Michel Marcus, Mar 10 2017

import math
f=math.factorial
def C(n, r): return f(n)/ f(r)/ f(n-r)
def A070307(n): return C(n + 2, 3)**n # Indranil Ghosh, Mar 10 2017

a(n) = C(n+2, 3)^n = A000292(n)^n.

a(n) ~ 6^(-n)*exp(3-5/(2*n))*n^(3*n). - Stefano Spezia, Jun 13 2025

More terms from Robert G. Wilson v, May 10 2002

More terms from Michel Marcus, Mar 10 2017

A331988 Table T(n,k) read by antidiagonals. T(n,k) is the maximum value of Product_{i=1..n} Sum_{j=1..k} r_j[i] where each r_j is a permutation of {1..n}.

Original entry on oeis.org

1, 2, 2, 6, 9, 3, 24, 64, 20, 4, 120, 625, 216, 36, 5, 720, 7776, 3136, 512, 56, 6, 5040, 117649, 59049, 10000, 1000, 81, 7, 40320, 2097152, 1331000, 248832, 24336, 1728, 110, 8, 362880, 43046721, 35831808, 7529536, 759375, 50625, 2744, 144, 9, 3628800, 1000000000, 1097199376, 268435456, 28652616, 1889568, 93636, 4096, 182, 10

Offset: 1

Chai Wah Wu, Feb 23 2020

A dual sequence to A260355. See arXiv link for sets of permutations that achieve the value of T(n,k). The minimum value of Product_{i=1..n} Sum_{j=1..k} r_j[i] is equal to n!*k^n.

			T(n,k)
   k    1    2     3      4      5      6      7      8      9     10     11     12
  ---------------------------------------------------------------------------------
n  1|   1    2     3      4      5      6      7      8      9     10     11     12
   2|   2    9    20     36     56     81    110    144    182    225    272    324
   3|   6   64   216    512   1000   1728   2744   4096   5832   8000  10648  13824
   4|  24  625  3136  10000  24336  50625  93636 160000 256036 390625 571536 810000

Chai Wah Wu, Table of n, a(n) for n = 1..70
Chai Wah Wu, Permutations r_j such that ∑i∏j r_j(i) is maximized or minimized, arXiv:1508.02934 [math.CO], 2015-2020.
Chai Wah Wu, On rearrangement inequalities for multiple sequences, arXiv:2002.10514 [math.CO], 2020.

Cf. A000027, A000142, A000169, A001504, A016743, A016766, A061711, A061718, A260355.

from itertools import permutations, combinations_with_replacement
def A331988(n,k): # compute T(n,k)
    if k == 1:
        count = 1
        for i in range(1,n):
            count *= i+1
        return count
    ntuple, count = tuple(range(1,n+1)), 0
    for s in combinations_with_replacement(permutations(ntuple,n),k-2):
        t = list(ntuple)
        for d in s:
            for i in range(n):
                t[i] += d[i]
        t.sort()
        w = 1
        for i in range(n):
            w *= (n-i)+t[i]
        if w > count:
            count = w
    return count

T(n,n) = (n*(n+1)/2)^n = A061718(n).
T(n,k) <= (k(n+1)/2)^n.
T(1,k) = k = A000027(k).
T(n,1) = n! = A000142(n).
T(2,2m) = 9m^2 = A016766(m).
T(2,2m+1) = (3m+1)*(3m+2) = A001504(m).
T(n,2) = (n+1)^n = A000169(n+1).
T(3,k) = 8k^3 = A016743(k) for k > 1.
If n divides k then T(n,k) = (k*(n+1)/2)^n.
If k is even then T(n,k) = (k*(n+1)/2)^n.
If n is odd and k >= n-1 then T(n,k) = (k*(n+1)/2)^n.
If n is even and k is odd such that k >= n-1, then T(n,k) = ((k^2*(n+1)^2-1)/4)^(n/2).

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A085741 a(n) = T(n)^n, where T() are the triangular numbers (A000217).

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A070307 Number of n X n matrices with nonnegative integer entries such that every row sum equals 3.

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A331988 Table T(n,k) read by antidiagonals. T(n,k) is the maximum value of Product_{i=1..n} Sum_{j=1..k} r_j[i] where each r_j is a permutation of {1..n}.

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